Journal ArticleDOI
Solving the nonlinear Poisson-type problems with F-Trefftz hybrid finite element model
TLDR
In this article, a hybrid finite element model based on F-Trefftz kernels (fundamental solutions) is formulated for analyzing Dirichlet problems associated with two-dimensional nonlinear Poisson-type equations.Abstract:
A hybrid finite element model based on F-Trefftz kernels (fundamental solutions) is formulated for analyzing Dirichlet problems associated with two-dimensional nonlinear Poisson-type equations including nonlinear Poisson–Boltzmann equation and diffusion–reaction equation. The nonlinear force term in the Poisson-type equation is frozen by introducing the imaginary terms at each Picard iteration step, and then the induced Poisson problem is solved by the present hybrid finite element model involving element boundary integrals only, coupling with the particular solution method with radial basis function interpolation. The numerical accuracy of the present method is investigated by numerical experiments for problems with complex geometry and various nonlinear force functions.read more
Citations
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Journal ArticleDOI
Method of fundamental solutions for nonlinear skin bioheat model
TL;DR: In this article, a two-dimensional nonlinear skin model with temperature-dependent blood perfusion rate was studied, where the original bioheat transfer governing equation was linearized with the Taylor's expansion technique, and the linearized governing equation with specified boundary conditions was solved using a meshless approach, in which the DRM and the MFS were employed to obtain particular and homogeneous solutions, respectively.
Journal ArticleDOI
A fundamental solution-based finite element model for analyzing multi-layer skin burn injury
Hui Wang,Qing-Hua Qin +1 more
TL;DR: In this paper, a fundamental solution-based hybrid finite element formulation is proposed for numerically simulating steady-state temperature distribution inside a multilayer human skin tissue during burning, since only element boundary integrals are involved, the computational dimension is reduced by one as the fundamental solutions used analytically satisfies the bioheat governing equation.
Journal ArticleDOI
A new meshless method for solving steady-state nonlinear heat conduction problems in arbitrary plane domain
TL;DR: In this paper, a multiple-scale polynomial expansion method was proposed to solve nonlinear heat conduction problems in arbitrary plane domains, where the multiple scales are automatically decided by the collocation points.
Journal ArticleDOI
A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients
Sergiy Reutskiy,Ji Lin +1 more
TL;DR: In this article, a semi-analytic numerical method for strongly nonlinear steady-state advection-diffusion-reaction equation (ADRE) in arbitrary 2D domains is presented.
Journal ArticleDOI
Method of particular solutions for nonlinear Poisson-type equations in irregular domains
TL;DR: In this article, the authors proposed a new meshless numerical technique for solving nonlinear Poisson-type equations in irregular domains using the dual reciprocity method (DRM) approach, where the nonlinear term is represented as a linear combination of basis functions.
References
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Book
The Trefftz Finite and Boundary Element Method
Qing-Hua Qin,VD Radulescu +1 more
TL;DR: In this paper, the concept of T-complete solution is used to compare T-elements with conventional finite elements with boundary elements, and a variational formulation for thin plate bending is proposed.
Journal ArticleDOI
A meshless local boundary integral equation (LBIE) method for solving nonlinear problems
T. Zhu,J. Zhang,Satya N. Atluri +2 more
TL;DR: In this paper, a meshless method for solving nonlinear boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation, is proposed.
Journal ArticleDOI
Newton iteration with multiquadrics for the solution of nonlinear PDEs
TL;DR: It is shown how globally supported multiquadric radial basis functions can be used for this task and one of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquADric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm.
Journal ArticleDOI
Particular solutions of Laplacian, Helmholtz-type, and polyharmonic operators involving higher order radial basis functions
TL;DR: Particular solutions of higher order radial basis functions of conical and spline types are obtained for the Laplacian, Helmholtz type, and polyharmonic operators in this paper.
Journal ArticleDOI
The method of fundamental solutions for non-linear thermal explosions
TL;DR: In this article, a numerical method based on the method of fundamental solutions, thin plate spine interpolation and monotone iteration is devised to find the minimal solution of the steady-state blow-up problem.
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